A Jarník type theorem for planar curves: everything about the parabola
A Jarník type theorem for planar curves: everything about the parabola
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DOI：
10.1017/s0305004115000195
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发表时间：
2013-09
影响因子：
0.8
通讯作者：
Mumtaz Hussain
中科院分区：
文献类型：
--
作者：
Mumtaz Hussain
摘要 度量丢番图近似中著名的 Khintchine 和 Jarník 定理提供了对给定误差下有理数可逼近的实数的测度理论性质的全面描述。对于其他设置，特别是对于曲线和更一般的流形，已经获得了这些基本结果的各种概括。在本文中，我们通过完善抛物线情况下的理论来发展平面曲线的理论。这是流形丢番图近似理论中同类研究中的首次综合研究。
Abstract The well-known theorems of Khintchine and Jarník in metric Diophantine approximation provide a comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.